Problem 66 - Algebra 2

Question

Determine whether each function has a maximum or a minimum value. Then find the maximum or minimum value of each function. $$ f(x)=x^{2}-8 x+3 $$

Step-by-Step Solution

Answer

The function has a minimum value of -13 at x = 4.

Step 1: Identify the Function Type

The given function is \[ f(x) = x^2 - 8x + 3 \].This is a quadratic function, and it can be written in the standard form of a quadratic equation \( ax^2 + bx + c \). Here, \( a = 1 \), \( b = -8 \), and \( c = 3 \).

Step 2: Determine if the Function has a Maximum or Minimum

For quadratic functions in the form \( ax^2 + bx + c \), the parabola opens upwards if \( a > 0 \) and downwards if \( a < 0 \). Since \( a = 1 \) which is greater than 0, the parabola opens upwards, meaning the function has a minimum value.

Step 3: Find the Vertex of the Parabola

The vertex form of a quadratic equation gives the minimum or maximum value. The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -8 \), so\[ x = -\frac{-8}{2 \times 1} = 4 \].

Step 4: Calculate the Minimum Value

Substitute \( x = 4 \) back into the function to find the minimum value.\[ f(4) = (4)^2 - 8 \times 4 + 3 \]\[ f(4) = 16 - 32 + 3 \]\[ f(4) = -13 \].Thus, the minimum value of the function is -13.

Key Concepts

Parabola

A parabola is a U-shaped curve that is the graphical representation of a quadratic function. Its general form embodies symmetry and can open either upwards or downwards depending on its leading coefficient. The direction it opens is determined by the sign of the coefficient "a" in the standard quadratic equation form: \( ax^2 + bx + c \).
If \( a > 0 \), the parabola opens upwards and this implies that the parabola has a minimum point. Conversely, if \( a < 0 \), it opens downwards, leading to a maximum point. Thus, in the context of the quadratic function \( f(x) = x^2 - 8x + 3 \), since \( a = 1 \) is positive, the parabola opens upwards and possesses a minimum value. This understanding of the parabola's direction is crucial in determining the nature of the function's extremum—whether it is a maximum or minimum value.

Vertex

In the context of a quadratic function, the vertex represents the highest or lowest point of the parabola. It is a significant characteristic because it indicates the parabola's extremum. The vertex can be located by transforming the quadratic equation into the vertex form \( f(x) = a(x-h)^2 + k \), where \((h, k)\) denotes the vertex coordinates.
Alternatively, for a function in standard form \( ax^2 + bx + c \), the x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \).

• Finding the vertex of \( f(x) = x^2 - 8x + 3 \):
  Here, \( a = 1 \) and \( b = -8 \) resulting in \( x = -\frac{-8}{2 \times 1} = 4 \).
  Substituting \( x = 4 \) into the original equation allows us to calculate the y-coordinate: \( f(4) = (4)^2 - 8 \times 4 + 3 = -13 \).

Thus, the vertex for this specific function is \((4, -13)\), marking the lowest point of the upwards-opening parabola.

Maximum and Minimum Values

The concepts of maximum and minimum values are central to understanding parabolas in quadratic functions. These values correspond to the highest or lowest points on the graph of the parabola. For quadratic functions, whether the extremum is a maximum or a minimum is dictated by the coefficient "a".
In the function \( f(x) = x^2 - 8x + 3 \):

• Since \( a = 1 \) is positive, the parabola opens upwards, indicating the presence of a minimum value at its lowest point, the vertex.
• The y-coordinate of the vertex, calculated previously as -13, represents this minimum value.

This characteristic reveals not only the function's behavior but also its critical point of turning, providing insight into its graphical nature and optimization capabilities.

Quadratic Equation Standard Form

The quadratic equation standard form is \( ax^2 + bx + c \), which is essential for analyzing quadratic functions. This form allows for straightforward identification of key elements such as the parabola's direction and vertex calculation. The coefficients \( a \), \( b \), and \( c \) serve distinct purposes:

• "a" indicates the direction of the parabola (upward for \( a > 0 \), downward for \( a < 0 \)) and affects the parabola's width.
• "b" influences the vertex's horizontal position and the symmetry of the graph.
• "c" represents the y-intercept, showing where the parabola intersects the y-axis.

Understanding the standard form facilitates the comparison of quadratic functions and helps in the systematic approach to solutions, as illustrated in the exercise where the function \( f(x) = x^2 - 8x + 3 \) is rendered in its most useful form to easily determine its key properties. By employing this form, you can quickly ascertain whether a function possesses a maximum or minimum value and compute its vertex efficiently.